Let $A \in \mathbb{R}^{n \times n}$ be a given symmetric and positive semidefinite matrix and consider the set $\Omega:=\left\{x \in \mathbb{R}^{n}: x^{\top} A x \leq 0\right\} .$ Show that the set $\Omega$ is convex.
My incomplete proof:
$\Omega:=\left\{x \in \mathbb{R}^{n}: x^{\top} A x = 0\right\} .$ Assume $y_1, \, y_2 \in \Omega$, prove that $\lambda y_1 + (1 - \lambda)y_2 \in \Omega$
Finally, I got $(\lambda y_1 + (1 - \lambda)y_2)^TA(\lambda y_1 + (1 - \lambda)y_2) = \lambda (1 - \lambda)(y_1^TAy_2 + y_2^TAy_1)$.
But I do not know what to do next. Can anybody help me?